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Stochastic programming problems with chance constraints
Harcek, Milan ; Branda, Martin (advisor) ; Kopa, Miloš (referee)
The thesis presents stochastic programming with chance contraints. We begin with the definition of convex set, convex and concave function and we study the convexity of programs with deterministic constraints. We continue with the definition of quasi-concave and quasi-convex function. After that, we put our mind to probabilistic constraints and the convexity of feasible set and show the formulation of joint and separate probabilistic constraints. We discuss properties of feasible set in general case, without any assumptions concerning the probability distribution of random variable. Finally, we apply our theory to random vectors with finite discrete distribution and multiva- riate normal distribution. 1
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Multivariate risk measures in stochastic optimization
Rauš, Jaroslav ; Branda, Martin (advisor) ; Dupačová, Jitka (referee)
The thesis deals with possible generalization of widely used risk measures, Value-at-Risk and Conditio- nal Value-at-Risk, to the multivariate case. First, the theory of p-efficient points, possible generalization of a quantile, is presented. The Prékopa-Vizvári-Badics algorithm for finding p-efficient points in case of random vectors with finite support is presented and a generalization of the algorithm in special case is proposed. Multivariate Value-at-Risk and Multivariate Conditional Value-at-Risk are defined and some of the properties are discussed. A lot-sizing problem for different time horizons is solved. 1
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Generalized Linear Models in Reserving Risk
Zboňáková, Lenka ; Pešta, Michal (advisor) ; Branda, Martin (referee)
In the presented thesis we deal with the generalized linear models framework in a claims reserving problem. Claims reserving in non-life insurance is firstly described and the considered class of models is introduced. Consequently, this branch of stochastic modelling is implemented in the reserving setup. For computation of the risk associated with claims reserving, we need a predictive distribution of future liabilities in order to evaluate risk measures such as Va- lue at Risk and Conditional Value at Risk. Since datasets in non-life insurance commonly consist of a small number of observations and estimation of predictive distributions can be complicated, we adopt a bootstrap method for this purpose. Model fitting, simulations and consequent measuring of the reserving risk are performed within the use of real-life data. Based on this, an analysis of fitted models and their comparison together with graphical outputs is included. 1
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Multivariate GARCH
Maďar, Milan ; Branda, Martin (referee) ; Mazurová, Lucie (referee)
4 Title: Multivariate GARCH Author: Mgr. Milan Mad'ar Department: Katedra pravděpodobnosti a matematické statistiky Abstract: This thesis will examine the regional and global linkages as evi- dence the integrated markets consist of stock markets in Frankfurt, Amsterdam, Prague the U.S. Therefore we will utilize the multivariate GARCH approach that investigates into the dynamics of volatility transmission of related foreign exchange rates. Also, we will define three basic model classes. For each of the model classes a theoretical review, basic properties and estimation procedure with proofs are provided. We illustrate approach by applying the models to daily market data. Our two main aims are discussing and reporting the existence of regional and global stock markets linkages and provide a comparison of such mul- tivariate GARCH models on the data sample. We find out that the estimated time-varying conditional correlations indicate limited integration among the mar- kets which implies that investors can benefit from the risk reduction by investing in the different stock markets especially during the crisis. Keywords: multivariate GARCH, VECH, BEKK, O-GARCH, GO-GARCH, CCC, DCC
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Sample approximation technique in stochastic programming
Vörös, Eszter ; Branda, Martin (advisor) ; Kozmík, Václav (referee)
Title: Sample approximation technique in stochastic programming Author: Eszter V¨or¨os Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Martin Branda, Ph.D., Department of Probability and Mathematical Statistics Abstract: This thesis deals with the problem of stochastic programming. Sto- chastic problems are usually applied for optimalization problems involving uncer- tain parameters. The problem, which we are aimed to solve, is approximated with the so-called sample average approximation method. The sample used to estimate the true problem is generated by the Monte Carlo method. This technique allows us to use standard algorithms for the further treatment of the problem. The aim of this thesis is to discuss the convergence properites of the optimal value and the optimal solution of the approximed problem to the optimal value and the optimal solution of the real problem. The thesis ends with a practical demonstration of the theoretical results on a portfolio optimization problem. Keywords: stochastic programming, sample average approximation, Monte Carlo method, portfolio optimization 1
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Bayesian Approaches to Stochastic Reserving
Novotová, Simona ; Pešta, Michal (advisor) ; Branda, Martin (referee)
In the master thesis the issue of bayesian approach to stochastic reserving is solved. Reserving problem is very discussed in insurance industry. The text introduces the basic actuarial notation and terminology and explains the bayesian inference in statistics and estimation. The main part of the thesis is framed by the description of the particular bayesian models. It is focused on the derivation of estimators for the reserves and ultimate claims. The aim of the thesis is to show the practical uses of the models and the relations between them. For this purpose the methods are applied on a real data set. Obtained results are summarized in tables and the comparison of the methods is provided. Finally the impact of a prior distribution on the resulting reserves is showed. Powered by TCPDF (www.tcpdf.org)
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Optimization problems of vertex coloring under uncertainty
Kučera, Petr ; Branda, Martin (advisor) ; Lachout, Petr (referee)
The goal of this paper is to compare two formulations of optimi- zation problems of vertex coloring under uncertainty. These two formulations are integer linear programming with constraints and integer quadratic without con- straints. First chapter introduces integer linear programming. In second chapter we learn about these two formulations. Third chapter deals with implementation of these two formulations in optimization program called GAMS. We randomly generate 20 optimization problems of vertex coloring under uncertainty and com- pare integer linear formulation with constraints and integer quadratic formulation without constraints. 1
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Optimal pricing in nonlife insurace
Ďurošková, Zuzana ; Branda, Martin (advisor) ; Mazurová, Lucie (referee)
V této práci se zabýváme základními principy tvorby sazeb neži- votního pojištění. Pracujeme s rizikově heterogenním portfóliem obsahující ur- čitý počet rizikových tříd. Cílem je najíst optimální sazbu pojistného pro kaž- dou třídu. K nalezení aplikujeme optimalizační modely a využíváme nelineárního programování. Formulujeme a řešíme optimalizační problém za jistých podmínek. Odvodíme jeho optimální řešení, z kterého vyjádříme a popíšeme r·zné principy pro výpočet pojistného pro každou třídu. Zavedeme taktéž duální optimalizační problém a ukážeme tvar jeho optimálního řešení. V numerické studii vypočítáme z odvozených metod sazby pojistného, kde pro jednotlivá rizika reprezentující úhrny škod, budeme volit konkrétní rozdělení. 1
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Optimal solutions and CLM sets
Semela, Ondřej ; Lachout, Petr (advisor) ; Branda, Martin (referee)
This thesis falls within the theory of optimization problems. In the first part, terms such as epi- convergence, lower and upper semicontinuous function, epi-continuity and CLM set are defined. For a better understanding, the definitions of the key terms are accompanied with illustrative examples and observations of their basic properties. The following part deals with searching of (local) minimizers of random or deterministic function. Using the knowledge from the first part it is showed that under a set of assumptions it is possible to transfer this search to a sequence of random functions of specific requirements. Powered by TCPDF (www.tcpdf.org)
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Exchange rate volatility, and central bank interventions
Kubů, Jan ; Lachout, Petr (advisor) ; Branda, Martin (referee)
The exchange rates of currencies of different countries show higher volatility than it could be explained by the volatility of the fundamental variables. There are introduced different models which try to describe the behavior of these exchange rates in this Diploma Thesis. Their comparison is made with respect to the ability to capture the volatility of the empirically observed data. The behavior of exchange rates may also be influenced by interventions of the state institutions and therefore we introduced models which allow the effect of such regulatory interventions. These models were applied on real data. The properties of the model predictions of exchange rates were compared and evaluated with respect to their ability to explain the volatility of the empirical data. At the summary of my work one of the models has been used to simulate the behavior of the exchange rate during the application of different intervention strategies of the Central Bank. Powered by TCPDF (www.tcpdf.org)
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